Date: Wed, 18 Aug 1999 09:26:22 +0200
From: "Gatherer, D. (Derek)" <D.Gatherer@organon.nhe.akzonobel.nl>
Subject: RE: Defection Rates and Classes (was Parody of Science)
To: "'memetics@mmu.ac.uk'" <memetics@mmu.ac.uk>
I'll start with a very extreme case, which is a society where there is no
movement between classes eg. D-up and D-down are zero. If Rp is 0.8 ie. 1.6
children per couple and Rw is 1.5 ie. 3 children per couple, we get:
cycle professionals workers %prof. total
0 100 900 10 1000
1 80 1350 5.59 1430
2 64 2025 3.06 2089
3 51 3038 1.66 3089
4 41 4556 0.89 4597
5 33 6834 0.48 6867
6 26 10252 0.26 10278
7 21 15377 0.14 15398
8 17 23066 0.07 23083
9 13 34599 0.04 34612
10 11 51899 0.02 51909
11 9 77848 0.01 77856
12 7 116772 0.01 116779
13 5 175158 0.00 175163
14 4 262736 0.00 262741
15 4 394105 0.00 394108
16 3 591157 0.00 591160
17 2 886735 0.00 886737
18 2 1330103 0.00 1330104
19 1 1995154 0.00 1995155
20 1 2992731 0.00 2992732
21 1 4489097 0.00 4489098
22 1 6733645 0.00 6733646
23 1 10100467 0.00 10100468
24 0 15150701 0.00 15150701
25 0 22726051 0.00 22726052
26 0 34089077 0.00 34089077
27 0 51133616 0.00 51133616
28 0 76700424 0.00 76700424
29 0 115050636 0.00 115050636
30 0 172575953 0.00 172575953
The average number of children a member of the professional classes might
expect is 1.6, and the average number of grandchildren is 1.6 squared =
2.56 and greatgrandchildren = 4.1. When we reach a level where 1.6 power n
>= the total prof. population, then any particular individual in generation
1 is likely to be the ancestor of the entire population. Here this happens
in generation 8, when there are only 17 professionals left.
This situation can occur in island isolates such as Ascension Island and the
Pitcairn Islands. Of course it's only an average estimate because variation
in numbers of progeny (especially amongst males who can have anything from
zero to dozens of children) means that in an isolated population one
original individual very rapidly can become the genetic 'founder' of the
population.
However, it's clear that this won't happen here because we can allow
migration into the professional class to keep its numbers up.
So if Dup = 5%
cycle prof workers %prof. total
0 100 900 10.00 1000
1 148 1283 10.31 1430
2 214 1828 10.49 2042
3 308 2604 10.59 2913
4 442 3711 10.64 4153
5 632 5288 10.67 5920
6 902 7536 10.69 8438
7 1287 10739 10.70 12026
8 1835 15302 10.71 17137
9 2616 21806 10.71 24422
10 3728 31074 10.71 34802
11 5313 44280 10.71 49593
12 7571 63099 10.71 70670
13 10789 89916 10.71 100705
14 15375 128130 10.71 143505
15 21910 182585 10.71 204495
16 31222 260184 10.71 291406
17 44491 370762 10.71 415253
18 63400 528336 10.71 591736
19 90345 752879 10.71 843224
20 1287421072852 10.71 1201594
so here the population is growing rapidly, and the 5% upwards migration rate
is just about enough to keep the proportions of professional to workers the
same. There is no extinction effect like we saw in the previous example.
However, for any _individual_ member of the professional classes in the
first generation the evolutionary prognosis is the same. Although the
professional class, as a class, holds its own, and even increases its
proportions slightly, most of the professional class is soon immigrants from
the workers. These immigrants too also acquire low reproductive behaviour
and they too leave little in the way of genes after a few generations. So
its a continual turnover of professionals, as each incoming wave of workers
adjusts its reproductive level downwards and genetically peters out.
But, what if we introduce D-down? Say, at only 0.1%
cycle prof workers %prof. total
0 100 900 10.00 1000
1 147 1283 10.31 1430
2 214 1828 10.48 2042
3 308 2605 10.58 2913
4 442 3712 10.63 4154
5 631 5291 10.66 5922
6 901 7540 10.68 8442
7 1286 10746 10.69 12032
8 1834 15314 10.69 17148
9 2614 21825 10.70 24439
10 3726 31103 10.70 34829
11 5311 44326 10.70 49637
12 7569 63171 10.70 70739
13 10787 90027 10.70 100814
14 15373 128301 10.70 143674
15 21908 182846 10.70 204755
16 31223 260581 10.70 291804
17 44497 371363 10.70 415860
18 63414 529244 10.70 592658
19 90374 754244 10.70 844618
20 1287951074901 10.70 1203696
Almost indistinguishable in terms of the overall class proportions and
distribution. It is however, very different in terms of the genetic
prospects of a single individual in the professional class. That individual
has an average of 1.6 children. However, some of those 1.6 children may
slip down into the worker class, and have a raised reproductive rate. So
that individual may have a larger average number of grandchildren than the
2.56 which he/she would have without any D-down permitted.
So, off the top of my head....
the average number of descendents of a person in prof. class at gen.1, at
cycle 2 is
[2*Rp]squared + (2*Rp * D-down * Rw) = 2.5696
ie. the number of descendents expected by reproduction in the professional
class plus the number of descendents in the workers.
Subsequently it may not be so simple, as some of the descendents who migrate
downwards will themselves have descendents who migrate back up, and who then
may have descendents who migrate back down etc..., so it may be recursive.
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